Website

Unique continuation for discrete harmonic functions. We discuss a surprising version of the classical Liouville theorem. It says that a harmonic function bounded on a large portion of the standard square lattice is bounded. The corresponding statement fails on higher dimensional lattices. The main result was obtained in a joint work with Lev Buhovsky, Alexander Logunov, and Mikhail Sodin. We will survey some applications, describe a recent interesting generalization by Bon-Rabee, Cooperman, and Ganguly, and list some open problems.

Remez-type inequality for solutions of elliptic equations. The classical Remez inequality for polynomials gives a precise bound on the supremum of a polynomial over an interval in terms of its supremum over a subset of the interval of fixed measure and the degree of the polynomial. We show that a similar result holds for solutions of elliptic PDEs, where the role of the degree is played by Almgren’s frequency function. The inequality has applications to nodal geometry of Laplace eigenfunctions. The talk is based on a joint work with Alexander Logunov.

Decay of solutions to real Schrodinger equations on the plane. We consider solutions of Schrödinger equations with bounded real-valued potentials. In the 1960s Landis asked if such solutions decay at most exponentially. We answer the question in dimension two, where one of the tools is a quasi-conformal change of variables. The talk is based on a joint work with Alexander Logunov, Nikolai Nadirashvili, and Fedya Nazarov.